Control of partial differential equations/Controllability of nonlinear control systems

In this section, we consider the controllability of nonlinear control systems modeled by nonlinear PDE's. There are no general methods to deal with this difficult problem. It is natural to first consider the problem of the controllability around an equilibrium of a nonlinear partial differential equation. Then the natural next step is to see if the linearized control system around this equilibrium is controllable. This is what we call the linear test in the following. We give an example of application below and mention methods and tools to deal with the cases where there is a problem of loss of derivatives.

Of course when the linearized control system is not controllable, one cannot say that the same non-controllability holds for the nonlinear system. Various methods can be used to deal with this case. Let us mention, in particular,

The linear test: the regular case

When the linearized control system around an equilibrium is controllable, one can try to use the inverse mapping theorem to get a local controllability result for the nonlinear control system. We illustrate this method on the Korteweg-de Vries nonlinear control system:
\[\tag{1}
y_t+y_x+y_{xxx}+yy_x=0, \, t\in(0,T), \, x\in (0,L),
\]

\[\tag{2}
y(t,0)=y(t,L)=0,\, y_x(t,L)=u(t),\, t\in(0,T).
\]

(Of course this is just an example: this method can be applied to many nonlinear PDE).
The linearized control system around \((y,u)=(0,0)\) is the control system
\[\tag{3}
y_t+y_x+y_{xxx}=0, \, x\in (0,L),\, t\in(0,T),
\]

\[\tag{4}
y(t,0)=y(t,L)=0,\, y_x(t,L)=u(t),\, t\in(0,T),
\]

where, at time \(t\ ,\) the control is \(u(t)\in \mathbb{R}\) and the state is \(y(t,\cdot):(0,L)\rightarrow \mathbb{R}\ .\) By [[../Examples_for_controllability_of_linear_control_systems#Theorem9|Theorem 9]], if
\[\tag{5}
L\not \in \mathcal{N}:=\left\{2\pi
\sqrt{\frac{j^2+l^2+jl}{3}};\, j,l\in \mathbb{N}\setminus\{0\}\right\},
\]

then, for every time \(T>0\ ,\) the control system ()-(4) is exactly controllable in time \(T\ .\) Hence one may expect that the nonlinear control system ()-(2) is at least locally controllable if (5) holds. The goal of this section is to prove that this is indeed true, a result due to Lionel Rosier (1997).

Well-posedness of the Cauchy problem

Let us first define the notion of solutions for the Cauchy problem associated with ()-(2). Multiplying () by \(\phi:[0,\tau]\times[0,L]\rightarrow \mathbb{R}\ ,\) using (2) and performing integrations by parts lead to the following definition.

is a function \(y\in C^0([0,T];L^2(0,L))\cap L^2((0,T);H^1(0,L))\) such that, for every \(\tau\in [0,T]\) and for every \(\phi \in C^3([0,\tau]\times[0,L])\) such that
\[\tag{9}
\phi(t,0)=\phi(t,L)=\phi_x(t,0)=0, \, \forall t\in[0,\tau],
\]

Theorem 11. Let \(T>0\ .\) Then there exists \(\varepsilon>0\) such that, for every \(y^0\in L^2(0,L)\) and \(u\in L^2(0,T)\) satisfying
\[
\|y^0\|_{L^2(0,L)} +\|u\|_{L^2(0,T)}\leqslant \varepsilon,
\]
the Cauchy problem ()-(7)-(8) has a unique solution. The proof is rather lengthy and technical. We omit it.

Local controllability

The goal of this section is to prove the following local controllability result (Lionel Rosier,1997).

Then there exist \(C>0\) and \(r_0>0\) such that for every \(y^0,y^1\in L^2(0,L)\ ,\) with \(\|y^0\|_{L^2(0,L)}<r_0\) and \(\|y^1\|_{L^2(0,L)}<r_0\ ,\) there exist
\[
y\in C^0([0,T],L^2(0,L))\cap L^2((0,T);H^1(0,L))
\]
and \(u\in L^2(0,T)\) satisfying ()-(2) such that
\[
y(0,\cdot)=y^0,
\]
\[
y(T,\cdot)=y^1,
\]
\[\tag{13}
\|u\|_{L^2(0,T)}\leqslant C (\|y^0\|_{L^2(0,L)}+\|y^1\|_{L^2(0,L)}).
\]

Sketch of a proof of Theorem 12. Let \(\mathcal{F}: L^2(0,L)\times L^2(0,T)\rightarrow L^2(0,L)^2\ ,\) \((y^0,u)\mapsto (y^0,y(T,\cdot))\) where \(y\in C^0([0,T];L^2(0,L))\) is the solution to the Cauchy problem ()-(7)-(8). It follows from the theorem in the previous subsection that \(\mathcal{F}\) is well defined on a neighborhood of\((0,0)\in L^2(0,L)\times L^2(0,T)\ .\) With some lengthy but straightforward estimates (see in particular Appendix A of (Coron and Crépeau, 2004), one can check that \(\mathcal{F}\) is of class \(C^1\) on a neighborhood of \((0,0)\in L^2(0,L)\times L^2(0,T)\) and that, as expected \(\mathcal{F}'(0,0): L^2(0,L)\times L^2(0,T)\rightarrow L^2(0,L)^2\ ,\) associates to \((y^0,u)\in L^2(0,L)\times L^2(0,T)(y^0,y(T,\cdot))\in L^2(0,L)^2\) where \(y\in C^0([0,T];L^2(0,L))\) is the solution to the Cauchy problem:
\[
y_t+y_x+y_{xxx}=0,\, t\in (0,T), \, x\in (0,L),
\]
\[
y(t,0)=y(t,L)=0,\, y_x(t,L)=u(t), \, t\in (0,T),
\]
\[
y(0,x)= y^0(x), \, x \in (0,L).
\]
From [[../Examples for Controllability of linear control systems#Theorem9|Theorem 9]], one gets that \(\mathcal{F}'(0,0)\) is onto, which by the inverse mapping theorem implies Theorem 12.

The linear test: the problem of loss of derivatives

In fact, in many situations one cannot apply directly the usual inverse mapping theorem to deduce from the controllability of the linearized control system at an equilibrium the local controllability of the nonlinear system at this equilibrium. This is due to some problem of loss of derivatives. Let us give a simple example where this problem appear. We consider the following simple nonlinear transport equation:
\[\tag{14}
y_t+a(y)y_x=0, \, x\in[0,L],\, t\in [0,T],
\]

\[\tag{15}
y(t,0)=u(t), \, t\in [0,T],
\]

where \(a \in C^2(\mathbb{R})\) satisfies
\[\tag{16}
0>
a(0)>0.
\]

For this control system, at time \(t\in [0,T]\ ,\) the state is \(y(t,\cdot)\in C^1([0,L])\) and the control is \(u(t)\in \mathbb{R}\ .\) If one wants to have a Hilbert space as a state space, one can also work with suitable Sobolev spaces (for example \(y(t,\cdot)\in H^2(0,L)\) is a suitable space). We are interested in the local controllability of the control system (14)-(15) at the equilibrium \((\bar y,\bar u)= (0,0)\ .\) Hence we first look at the linearized control system at the equilibrium \((\bar y ,\bar u)= (0,0)\ .\) This linear control system is the following one:
\[\tag{17}
y_t+a(0)y_x=0, \, t\in [0,T],\, x\in[0,L],
\]

\[\tag{18}
y(t,0)=u(t), \, t\in [0,T].
\]

For this linear control system, at time \(t\in [0,T]\ ,\) the state is \(y(t,\cdot)\in C^1([0,L])\) and the control is \(u(t)\in \mathbb{R}\ .\) Concerning the well-posedness of the Cauchy problem of this linear control system, one easily gets the following proposition.

Of course (19) is a consequence of (22) and (23): it is a necessary condition for
the existence of a solution \(y\in C^0([0,T]\times[0,L])\) to the Cauchy problem (21)-(22)-(23). Similarly (20) is a direct consequence of (21), (22) and (23): it is a necessary condition for the existence of a solution \(y\in C^1([0,T]\times[0,L])\) to the Cauchy problem (21)-(22)-(23).

One has the following (easy) proposition.

Proposition. Let \(T>L/a(0)\ .\) The linear control system (17)-(18) is controllable in time \(T\ .\) In other words, for every \(y^0\in C^1([0,L])\) and for every \(y^1\in C^1([0,L])\ ,\) there exists \(u\in C^1([0,T])\) such that the solution \(y\) of the Cauchy problem (21)-(22)-(23) satisfies
\[\tag{24}
y(T,x)=y^1(x), \, x \in [0,L].
\]

In fact one can give an explicit example of such a \(u\ .\) Let \(u\in C^1([0,T])\) be such that
\[\tag{25}
u(t)=y^1(a(0)(T-t)), \text{ for every } t\in [T-(L/a(0)),T],
\]

\[\tag{26}
u(0)=y^0(0),
\]

\[\tag{27}
\dot u(0)=-a(0)y^0_x(0).
\]

Such a \(u\) exists since \(T>L/a(0)\ .\) Then the solution \(y\) of the Cauchy problem (21)-(22)-(23) is given by
\[\tag{28}
y(t,x)=y^0(x-a(0)t), \, \forall (t,x)\in[0,T]\times [0,L] \text{ such that } a(0)t\leqslant x,
\]

\[
y(t,x)=u(t-(x/a(0))), \, \forall (t,x)\in[0,T]\times [0,L] \text{ such that } a(0)t> x.
\]
(The fact that such a \(y\) is in \(C^1([0,T]\times[0,L])\) follows from the fact that \(y\) and \(u\) are of class \(C^1\) and from the compatibility conditions ()-().) From (28), one has (23). From \(T>L/a(0)\ ,\) () and (28), one gets
(24). With this method, one can easily construct a continuous linear map:
\[
\Gamma:C^1([0,L])\times C^1([0,L]) \rightarrow C^1([0,T])
\]
\[
(y^0,y^1)\mapsto u
\]
such that

In order to prove the local controllability of the control system (14)-(15) at the equilibrium \((\bar y,\bar u) = (0,0)\ ,\) let us try to mimic what we have done for the nonlinear Korteweg-de Vries equation ()-(2). Now the map \(\mathcal{F}\) is the following one. Let
\[
F:=\{(y^0,u)\in C^1([0,L])\times C^1([0,T]);\, u(0)=y^0(0), \, \dot u(0)+a(y^0(0))y_x^0(0)=0\},
\]
\[
G:= C^1([0,L])^2.
\]
\[
\mathcal{F}:F\rightarrow G
\]
\[
(y^0,u)\mapsto (y^0,y(T,\cdot))
\]
where \(y\in C^1 ([0,T]\times[0,L]))\) is the solution to the Cauchy problem:
\[\tag{29}
y_t+a(y)y_x=0, \, x\in[0,L],\, t\in [0,T],
\]

\[\tag{30}
y(t,0)=u(t), \, t\in [0,T],
\]

\[\tag{31}
y(0,x)=y^0(x), \forall x\in [0,L].
\]

One can prove that this map \(\mathcal{F}\) is well defined and continuous in a neighborhood of \((0,0)\) (see (Li and Yu, 1985) for much more general result). Note that, formally, the linearized control system at \((y,u):=(0,0)\) is the control system (17)-(18), which by above proposition is controllable (at least if \(T>L/a(0)\)). Unfortunately the map \(\mathcal{F}\) is not of class \(C^1\ .\) One could think to avoid this problem by replacing \(G\) by \(G:= C^1([0,L])\times C^0([0,T])\ .\) Then the map \(\mathcal{F}\) is now of class \(C^1\ ,\) but \(\mathcal{F}'(0,0):F\rightarrow G\) is no longer onto: there are controls \(u\in C^0([0,T])\) allowing to go from \(y^0\in C^1([0,L]\) to \(y^1\in C^0([0,L])\) but these controls are not of class \(C^1\) if \(y^0\) is not of class \(C^1\ .\) We have lost one derivative.

1. It does not give the optimal functional spaces for the state and the control.

2. It is more complicated to apply than the method we want to present here.

There is a more standard fixed point method which works for many control systems, in particular for the control system (17)-(18) which allows to prove for this control system the following controllability result.

Theorem 13. Let us assume that
\[\tag{32}
T>\frac{L}{a(0)}.
\]

Then there exist \(\varepsilon>0\) and \(C>0\) such that, for every \(y^0\in C^1([0,L])\) and for every \(y^1\in C^1([0,L])\) such that
\[\|y^0\|_{C^1([0,L])}\leqslant \varepsilon \text{ and }\|y^1\|_{C^1([0,L])}\leqslant \varepsilon,
\]
there exists \(u\in C^1([0,T])\) such that
\[\tag{33}
\|u\|_{C^1([0,T])}\leqslant C(\|y^0\|_{C^1([0,L])}+\|y^1\|_{C^1([0,L])})
\]

and such that the solution of the Cauchy problem (29)-(30)-(31) exists, is of class \(C^1\) on \([0,T]\times[0,L]\) and satisfies
\[
y(T,x)=y^1(x),\, x \in[0,L].
\]

The fixed point method is the following one. Let \(z\in C^1([0,L]\times [0,L]\ .\) One consider the following linear control system
\[\tag{34}
y_t+a(z)y_x=0,\, y(t,0)=u(t), \, t\in [0,T],\, x\in [0,L].
\]

If the \(C^1\)-norm of \(z\) is small enough this system is controllable in time \(T>L/a(0)\) and there exist \(u\in C^1([0,T])\) such that the Cauchy problem
\[
y_t+a(z)y_x=0,\, y(t,0)=u(t),\, y(0,x)=y^0(x), \, t\in [0,T],\, x\in [0,L],
\]
has a (unique) solution \(y\in C^1([0,T]\times[0,L])\) and this solution satisfies \(y(T,x)=y^1(x), \, \forall x \in [0,L]\ .\) Of course this \(u\) is not unique. However, if one chooses it well, one can prove, using the Brouwer fixed point theorem, that, at least if the \(C^1\)-norm of \(y^0\) and \(y^1\) are small enough, the map \(z\mapsto y\) has a fixed point, which shows that the control \(u:=y(\cdot,0)\) steers the control system (14)-(15) from \(y^0\) to \(y^1\ .\) See Section 4.2 in (Jean-Michel Coron, 2007) for more details.

Remark. As for the Nash-Moser method, the controllability of the linearized control system (21)-(22)-(23) at the equilibrium \((y ,u):=(0,0)\) is not sufficient for our proof of the above theorem: one needs a controllability result for linear control systems which are close to the linear control system
(21)-(22)-(23).

Remark. Sometimes these fixed point methods used with careful estimates can lead to global controllability results if the nonlinearity is not too strong at infinity. See in particular

The return method

In order to explain this method, let us first consider the problem of local controllability of the following control system in finite dimension
\[
\dot y = f(y,u),
\]
where \(y\in\mathbb{R}^{n}\) is the state and \(u\in\mathbb{R}^{m}\) is the control; we assume that \(f\) is of class \(C^\infty\) and satisfies
\[
f(0,0)=0.
\]
The return method consists in reducing the local controllability of a nonlinear control system to the existence of suitable trajectories and to the controllability of linear systems. The idea is the following one: Assume that, for every positive real number \(T\) and every positive real number \(\varepsilon\ ,\) there exists a measurable bounded function \(\bar
u:[0,T]\rightarrow\mathbb{R}^{m}\) with \(\|\bar u\|_{L^\infty(0,T)}\leqslant \varepsilon\) such that, if we denote by \(\bar y\) the solution of \(\dot{\bar y} =f(\bar y,\bar u(t))\ ,\) \(\bar y(0)=0\ ,\) then
\[\tag{35}
\bar y(T)=0,
\]

Then, from the inverse function theorem, one gets the existence of \(\eta>0\) such that, for every \(y^0\in\mathbb{R}^n\) and for every \(y^1\in \mathbb{R}^n\) satisfying
\[
|y^0|< \eta , \, |y^1|< \eta,
\]
there exists \(u\in L^\infty((0,T);\mathbb{R}^m)\) such that
\[
|u(t)-\bar u(t)|\leqslant \varepsilon, \, t\in [0,T],
\]
and such that, if \(y:[0,T]\rightarrow \mathbb{R}^n\) is the solution of the Cauchy problem
\[
\dot y =f(y,u(t)), \, y(0)=y^0,
\]
then
\[
y(T)=y^1.
\]
Since \(T>0\) and \(\varepsilon >0\) are arbitrary, one gets that \(\dot y=f(y,u)\) is small-time locally controllable at the equilibrium \((0,0)\in \mathbb{R}^n\times\mathbb{R}^m\ .\)

where the state is \(y=(y_1,y_2,y_3)^\text{tr} \in \mathbb{R}^3\) and the control is \(u=(u_1,u_2)^\text{tr}\in \mathbb{R}^2.\) Let us recall that this system is small-time locally controllable at \((0,0) \in \mathbb{R}^3\times \mathbb{R}^2\ .\) The classical proof of this property relies on Lie brackets and on the Rashevski-Chow theorem (Petr K. Rashevski, 1938)-(Wei-Liang Chow, 1939); see e.g. Theorem 3.19 page 135 in (Jean-Michel Coron, 2007). Let us show how the return method also gives this controllability property. Take any \(T>0\) and any \(\bar u :[0,T]\rightarrow \mathbb{R}^2\) such that \(\bar u(t-t)=-\bar u(t)\ .\) Let \(\bar y :[0,T]\rightarrow \mathbb{R}^3\) be the solution of the Cauchy problem
\[
\dot {\bar y}=f(\bar y, \bar u(t)), \, \bar y(0)=0.
\]
One easily checks that \(\bar y(T)=0\) and that the linearized control system around \((\bar y, \bar u)\) is controllable if (and only if) \(\bar u\not\equiv 0\ .\) Hence we recover the small-time local controllability at \((0,0) \in \mathbb{R}^3\times \mathbb{R}^2\ .\) Let us point out that this new proof does not use any Lie bracket and uses only controllability of linear control systems. This is exactly what we want in order to deal with nonlinear control systems modeled by partial differential equations: For these systems, Lie brackets often do not work and one knows a lot of tools to study the controllability of linear control systems modeled by partial differential equations (see, e.g., at this link).

With this return method, in (Jean-Michel Coron, 1993; 1996) and in (Olivier Glass, 1997; 2000) the following global controllability result of the Euler equations of incompressible fluids is proved (where, for simplicity we do not specify the regularity of the functions and of the domain \(\Omega\)):

Main ingredients for the proof of Theorem 14. Note that the linearized control system of the Euler control system is far from being controllable (for this linear system the vorticity cannot be modified). The proof relies on the return method. In order to use this method, one needs to construct a (good) trajectory \(\bar y\) going from \(0\) to \(0\ .\) Such a trajectory is constructed using a potential flow (i.e. a flow of the form \(\bar y(t,x)=\nabla \varphi (t,x)\)). If the potential flow is well chosen the linearized control system around this trajectory is controllable. Using this controllability and a suitable fixed point argument one gets the local controllability of the Euler control system. The global controllability follows from this local result by a suitable scaling argument.

Remark. The return method has been introduced in (Jean-Michel Coron, 1992) for a stabilization problem. It has been used for the first time in (Jean-Michel Coron, 1993; 1996) for the controllability of a partial differential equation, namely the Euler equations of incompressible fluids. The return method has been used to study the controllability
of the following partial differential equations:

Remark. In fact, as already mentioned in the previous section, for many nonlinear partial differential equations, the fact that the linearized control system along the trajectory \((\bar y,\bar u)\) is controllable is not sufficient
to get the local controllability along \((\bar y,\bar u)\ .\) This is due to some loss of derivatives problems. To take care of this problem, one uses some suitable fixed point methods. Note that these fixed point methods rely often on the controllability
of some (and many) linear control systems which are not the linearized control system along the trajectory \((\bar y, \bar u)\ .\) (It can also rely on some specific methods which do not use the controllability of any linear control system. This last case appears in the papers (Thierry Horsin, 1998) and (Olivier Glass, 2006).)

Remark. For the Navier-Stokes control system, the linearized control system around \(0\) is controllable, which leads to local controllability results for the Navier-Stokes control system itself; see (Oleg Imanuvilov, 1998; 2001) and (Enrique Fernàndez-Cara, Sergio Guerrero, Oleg Imanuvilov and Jean-Pierre Puel, 2004). However it is not clear how to deduce from this controllability a global controllability result for the Navier-Stokes control system. Roughly speaking, the global controllability results for the Navier-Stokes equations in (Jean-Michel Coron, 1995/96) and (Jean-Michel Coron and Andrei Fursikov, 1996) are deduced from the controllability of the Euler equations. This is possible because the Euler equations are quadratic and the Navier-Stokes equations are the Euler equations plus a linear "perturbation" (of course some technical problems appear due to the fact that this linear perturbation involves more derivatives than the Euler equations: One faces a problem of singular perturbations). See also the Section 3.5.2 in (Jean-Michel Coron, 2007).

Quasi-static deformations

Let us explain how this method can be used on a specific example. We consider the water-tank control system
\[\tag{38}
H_t\left(t,x\right)+(Hv)_x\left(t,x\right)=0, \, x\in[0,L],
\]

This is a control system, denoted \(\Sigma\ ,\) where, at time \(t\in[0,T]\ ,\)

the state is \(Y(t)=(H(t,\cdot),v(t,\cdot),s(t),D(t))\ ,\)

the control is \(u(t)\in \mathbb{R}\ .\)

Of course, the total mass of the fluid is conserved so that, for every solution of (38) to (40),
\[\tag{43}
\frac{\text{d}}{\text{d} t}\int_0^L H\left(t,x\right) \text{d}x=0.
\]

(One gets (43) by integrating (38) on \([0,L]\) and by using (40) together with an integration by parts.) Moreover, if \(H\) and \(v\) are of class \(C^1\ ,\) it follows from (39) and (40) that
\[\tag{44}
H_x(t,0)=H_x(t,L),
\]

which is also \(-u\left(t\right)/g\ .\) Therefore we introduce the vector space \(E\) of functions \( Y=(H,v,s,D)\in C^1([0,L])\times C^1([0,L])\times\mathbb{R} \times\mathbb{R}\) such that
\[\tag{45}
H_x(0)=H_x(L),
\]

The vector space\(E\)is equipped with the natural norm
\[
|Y|:=\left\|H\right\|_{C^1([0,L])}+\left\|v\right\|_{C^1([0,L])}+|s|+|D|.
\]
One has the following local controllability theorem (Jean-Michel Coron, 2002).

As a corollary of this theorem, any steady state \(Y^1=(H_e,0,0,D^1)\) can be reached from any other
steady state \(Y^0=(H_e,0,0,D^0)\ .\)

Main ideas of the proof of Theorem 15. For simplicity, let us forget around the variables \(s\) and \(D\ .\) Without loss of generality, we may assume that \(H_e=g=L=1\ .\) Again the linearized control system around \((H,v,u):=(1,0,0)\) is not controllable. Indeed, this linearized control system is
\[
\Sigma_{\text{lin}}\quad \quad h_t+v_x=0,\, v_t+h_x=-u\left(t\right),\, v(t,0)=v(t,1)=0,
\]
and, if we let
\[
v_{\text{o}}(t,x):=\frac{1}{2}(v(t,x)-v(t,1-x)),\,h_{\text{e}}(t,x):=\frac{1}{2}(h(t,x)+h(t,1-x)),\,
\]
one gets
\[
h_{\text{e}t}+v_{\text{o}x}=0,\, v_{\text{o}t}+h_{\text{e}x}=0, v_{\text{o}}(t,0)=v_{\text{o}}(t,1)=0.
\]
Hence the control has no effect on \((h_{\text{e}},v_{\text{o}})\ ,\) which shows that \(\Sigma_{\text{lin}}\) is far from being controllable (it misses an infinite dimensional space).

Again, one tries to use the return method (see the previous section). For this method one needs to find trajectories such that the linearized control system around this trajectory is controllable. One can check that this is the case for the trajectory given by the following equilibrium point
\[
H_\gamma (x):= 1+\gamma (1/2-x), \, v_\gamma(x):=0, \, u_\gamma :=\gamma .
\]
where \(\gamma \in \mathbb{R}\) is small but not \(0\ .\) However this trajectory does not go from \(0\) to \(0\ :\) From the controllability of the linearized control system around \((H_\gamma, v_\gamma, u_\gamma)\) one gets only the local controllability around \((H_\gamma, v_\gamma)\ ,\) i.e., there exists an open neighborhood \(\mathcal{N}\) of \((H_\gamma, v_\gamma)\) such that, given two states in \(\mathcal{N}\ ,\) there exists a trajectory of the control system \(\Sigma\) going from the first state to the secund one. This does not imply the local controllability around \((1,0)\ .\) However, let us assume that

(i) There exists a trajectory of the control system \(\Sigma\) going from \((1,0)\) to \(\mathcal{N}\ ,\)

(ii) There exists a trajectory of the control system \(\Sigma\) going from some point in \(\mathcal{N}\) to \((1,0)\ .\)

Then it is not hard to prove the desired local controllability around \((1,0)\ .\) In order to get the trajectory mentioned in (i) and (ii), one uses quasi-static deformations: For example, for (i), one fixes some functions \(g:[0,1]\rightarrow \mathbb{R}\) such that \(g(0)=0,\, g(1)=1\) and consider for \(\varepsilon\in (0,+\infty)\) the control \(u_\varepsilon:[0,1/\varepsilon]\rightarrow \mathbb{R}\) defined by
\[
u_\varepsilon(t):= g(\varepsilon t).
\]
Let us now start from \((1,0)\) and use the control \(u_\varepsilon\ .\) Then one can check that, uniformly in \(t\in [0,1/\varepsilon]\ ,\)
\[
H(t,\cdot)-H_{g(\varepsilon t)}\rightarrow 0 \text{ and } v(t,\cdot)\rightarrow 0 \text{ as } \varepsilon \rightarrow 0.
\]
In particular \((H(1/\varepsilon ,\cdot),v(t,\cdot)\) in \(\mathcal{N}\) for \(\varepsilon \in (0,+\infty)\) small enough. This proves \((i)\ .\) The proof of \((ii)\) is similar.

Remark. There is in fact a problem of loss of derivatives and it is not easy to deduce the local controllability around \((H_\gamma, v_\gamma,u_\gamma)\) from the controllability of the linearized control system around \((H_\gamma, v_\gamma,u_\gamma)\ .\) To avoid the Nash-Moser method, a suitable fixed point method is used. This method requires the controllability "many" (essentially a family of codimension 4)) linear control systems which are close to the linear control system \(\Sigma_{\text{lin}}\ .\)

Remark. Note that, due to the finite speed of propagation, it is natural that only large-time local controllability holds.
However, for a Schrödinger analog of control system \(\Sigma\ ,\) it is proved in (Jean-Michel Coron, 2006) (see also Remark 9.20, pages 269-270 in (Jean-Michel Coron, 2007)) that the small-time local controllability also does not hold, even if the Schrödinger equation has an infinite speed of propagation. (The proof of the large-time local controllability for this Schrödinger control system is in (Karine Beauchard, 2005); see also (Karine Beauchard and Jean-Michel Coron, 2006) if one deals also with the variables \(S\) and \(D\ .\))

Power series expansion

Again, we present this method on an example. Let\(L>0\ .\) Let us consider the following Korteweg-de Vries control system
\[\tag{49}
y_t+y_x+y_{xxx}+yy_x=0, \, t\in(0,T), \, x\in (0,L),
\]

\[\tag{50}
y(t,0)=y(t,L)=0,\, y_x(t,L)=u(t),\, t\in(0,T),
\]

where, at time \(t\in [0,T]\ ,\) the control is \(u(t)\in \mathbb{R}\) and the state is \(y(t,\cdot):(0,L)\rightarrow \mathbb{R}\ .\)

We are interested in the local controllability of the control system ()-(50) around the equilibrium \((y_e,u_e):=(0,0)\ .\) Lionel Rosier has proved in (Rosier, 1997) that this local controllability holds if
\[\tag{51}
L\notin \mathcal{N}:=\left\{2\pi\sqrt{\frac{j^2+l^2+jl}{3}};\, j\in \mathbb{N}\setminus\{0\},\, l\in \mathbb{N}\setminus\{0\}\right\}.
\]

Lionel Rosier has got his result by first proving that, if \(L\not \in \mathcal{N}\ ,\) then the linearized control system around \((y_e,u_e):=(0,0)\) is controllable. Note that \(2\pi \in \mathcal{N}\) (take \(j=l=1\)) and, as shown also by Lionel Rosier, if \(L \in \mathcal{N}\ ,\) then the linearized control system around \((y_e,u_e):=(0,0)\) is not controllable. However the nonlinear term \(yy_x\) helps to recover the local controllability: One has the following theorem (Jean-Michel Coron and Emmanuelle Crépeau, 2004).

Theorem 16. Let \(T>0\) and let \(L=2\pi\ .\) (Thus, in particular, \(L\in\mathcal{N}\ .\)) Then there exist \(C>0\) and \(r_1>0\) such that for any \(y^0,\, y^1\in L^2(0,L)\ ,\) with \(\|y^0\|_{L^2(0,L)}<r_1\) and \(\|y^1\|_{L^2(0,L)}<r_1\ ,\) there exist
\[
y\in C^0([0,T];L^2(0,L))\cap L^2((0,T);H^1(0,L))
\]
and \(u\in L^2(0,T)\) satisfying ()-(50), such that
\[\tag{52}
y(0,\cdot)=y^0,
\]

Main ideas of the proof of Theorem 16. The proof relies on some kind of power series expansion. Let us just explain the method on the control system of finite dimension
\[\tag{55}
\dot y =f(y,u),
\]

where the state is \(y\in \mathbb{R}^n\) and the control is \(u\in \mathbb{R}^m\ .\) Here \(f\) is a function of class \(C^\infty\) on a neighborhood of \((0,0)\in \mathbb{R}^n\times\mathbb{R}^m\) and we assume that
\((0,0)\in \mathbb{R}^n\times\mathbb{R}^m\) is an equilibrium of the control system (55), i.e \(f(0,0)=0\ .\) Let
\[
H:=\text{Span} \,\{A^iBu;\, u\in \mathbb{R}^m,\,i\in \{0,\ldots,n-1\}\}
\]
with
\[
A:=\frac{\partial f}{\partial y}(0,0),\, B:= \frac{\partial f}{\partial u}(0,0).
\]
If \(H=\mathbb{R}^n\ ,\) the linearized control system around \((0,0)\) is controllable and therefore the nonlinear control system (55) is small-time locally controllable at \((0,0)\in \mathbb{R}^n\times\mathbb{R}^m\ .\) Let us look at the case where the dimension of \(H\) is \(n-1\ .\) Let us make a (formal) power series expansion of the
control system (55) in \((y,u)\) around the constant trajectory \(t\mapsto (0,0)\in \mathbb{R}^n\times\mathbb{R}^m\ .\) We write
\[
y=y^1+y^2+\ldots,\, u=u^1+u^2+\ldots \,.
\]
The order 1 is given by \((y^1,u^1)\ ;\) the order 2 is given by \((y^2,u^2)\) and so on. The dynamics of these different orders are given by
\[\tag{56}
\dot y^1=\frac{\partial f}{\partial y}(0,0)y^1+\frac{\partial f}{\partial u}(0,0)u^1,
\]

Hence, using the Brouwer fixed-point theorem and standard estimates on ordinary differential equations, one gets the local controllability of \(\dot y =f(y,u)\) (around \((0,0)\in \mathbb{R}^n\times\mathbb{R}^m\)) in time \(T\ ,\)
that is, for every \(\varepsilon>0\ ,\) there exists \(\eta>0\) such that, for every \((a,b)\in \mathbb{R}^n\times\mathbb{R}^n\) with \(|a|<\eta\) and \(|b|<\eta\ ,\) there exists a trajectory \((y,u):[0,T]\rightarrow \mathbb{R}^n\times\mathbb{R}^m\) of the control system (55) such that
\[
y(0)=a, \, y(T)=b,
\]
\[
|u(t)|\leqslant \varepsilon , \, t\in (0,T).
\]
We use this power series expansion method to prove the theorem. In fact, for this theorem, an expansion to order 2 is not sufficient: we obtain the local controllability by means of an expansion up to order 3 (which makes the computations very lengthy).

Remark. The power series expansion method has been used in the context of partial differential equations for the first time in (Jean-Michel Coron and Emmanuelle Crépeau, 2003). It has then been used